Question

A hemispherical bowl of radius R is rotated about its axis of symmetry which is kept vertical. A small block is kept in the bowl at a position where the radius makes an angle $\theta$ with the vertical. The block rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is $\mu$. Find the range of the angular speed for which the block will not slip.

Answer 1

Range of angular speed is $\left[ \sqrt{\frac{g}{R\sin\theta} \frac{\tan\theta - \mu}{1 + \mu\tan\theta}}, \sqrt{\frac{g}{R\sin\theta} \frac{\tan\theta + \mu}{1 - \mu\tan\theta}} \right]$ with appropriate conditions for $\mu$ and $\theta$.

Answer 2

The problem involves finding the range of angular speeds for which a block in a rotating hemispherical bowl does not slip. This is determined by analyzing the forces acting on the block at the minimum and maximum angular speeds, considering static friction.

Case 1: Minimum Angular Speed ($\omega_{min}$)

At this speed, the block tends to slip downwards, so friction acts upwards. We resolve forces into horizontal and vertical components to find the normal force $N$ and then $\omega_{min}$.

Vertical Equilibrium: $N\cos\theta + f_s\sin\theta - mg = 0$

Horizontal Motion (Centripetal Force): $N\sin\theta - f_s\cos\theta = m\omega_{min}^2 R\sin\theta$

Solving these equations, we find: $\omega_{min} = \sqrt{\frac{g}{R\sin\theta} \frac{\tan\theta - \mu}{1 + \mu\tan\theta}}$ (for $\tan\theta \ge \mu$)

If $\tan\theta < \mu$, $\omega_{min} = 0$.

Case 2: Maximum Angular Speed ($\omega_{max}$)

At this speed, the block tends to slip upwards, so friction acts downwards. We again resolve forces into horizontal and vertical components.

Vertical Equilibrium: $N\cos\theta - f_s\sin\theta - mg = 0$

Horizontal Motion (Centripetal Force): $N\sin\theta + f_s\cos\theta = m\omega_{max}^2 R\sin\theta$

Solving these equations, we find: $\omega_{max} = \sqrt{\frac{g}{R\sin\theta} \frac{\tan\theta + \mu}{1 - \mu\tan\theta}}$ (for $\tan\theta < 1/\mu$)

If $\tan\theta \ge 1/\mu$, $\omega_{max} = \infty$.

Therefore, the range of angular speed is $[\omega_{min}, \omega_{max}]$.